Development of a dual-mode inertial shear actuator for active vibration control of gear transmission systems

Abstract

In this paper, a new inertial mass actuator concept is presented for an active vibration control system. The actuator comprises two shear actuators and an inertial mass. The primary objective of the proposed actuator is to mitigate the vibration levels associated with the high-frequency local bending modes, present in the housing of gear transmission systems. Additionally, the actuator is constructed to be integrated into the gearbox housing. The actuator targets the first and second local deflection modes at the critical vibrating points of the gearbox housing. Moreover, the new actuator is designed to be energy efficient due to its low electrical capacitance. The concept of the proposed inertial mass shear actuator is introduced and a model is built to test the effectiveness of the active vibration control system. The concept is tested on a basic plate structure that has properties similar to the housing of the transmission system, and the developed actuator was able to reduce the vibration levels significantly. Approximately 36 dB reduction is achieved at steady state when targeting the first bending mode and approximately 41 dB is achieved at steady state by targeting the second bending mode.

1. Introduction

Regulating carbon emissions is a critical challenge that is confronting the global community. Greenhouse gas concentrations lead to global warming and climate disruptions. Transportation vehicles are major contributors to carbon emissions, prompting widespread regulatory efforts aimed at reducing them. In addition, the world is facing great challenges in the energy sector, so a huge effort is made to enhance the vehicle’s energy efficiency.

Reducing vehicle weight is crucial for improving fuel efficiency and reducing carbon emissions. One of the main components that contribute to the weight of the vehicle is the transmission system. However, reducing the transmission system’s weight can compromise its structural stiffness, leading to increased noise and vibration levels. Vehicle manufacturers face challenges due to the existence of regulations concerning Noise, Vibration, and Harshness (NVH) that determine the permissible levels of noise and vibration produced by the vehicles.

Several passive methods are used to mitigate the vibration and noise emissions in automobiles. However, Active Vibration Control (AVC) systems are exceptionally effective, they provide real-time adaptability and precision. Numerous academic studies have extensively investigated AVC in gearboxes and have highlighted significant advances in the field. For instance, Chen and Brennan (2000) developed an AVC system for a single-stage gearbox, they addressed directly the gear vibrations and used three magnetostrictive actuators fitted on the targeted gear. A feed-forward control system was utilized to determine the required control forces. They achieved 7 dB of reduction in gear angular vibrations at the gear meshing frequencies. Benzel and Möckel (2014, 2015) introduced an AVC system that utilizes an electronically commutated motor as an actuator to suppress gear vibrations. They proposed an adaptive feed-forward controller based on the Filtered-x Least Means Squares (FxLMS), and they achieved a housing vibration reduction of 31 dB and an airborne sound reduction of 13 dB at the first gear mesh frequency. Zhang et al. (2022) introduced a variable-step-size multichannel FxLMS algorithm. They established an experimental setup for active vibration control of a multi-stage gearbox, employing a piezoelectric stack actuator mounted on an auxiliary gear shaft. The vibration amplitude decreased by 3–9 dB at the designated point, while achieving an enhancement of 23–123% in the control precision. Wang et al. (2017, 2018) implemented an AVC system using two piezoelectric actuators, one mounted on the low-speed shaft and the other on the high-speed shaft of a multi-stage gearbox. They proposed an adaptive fuzzy proportion integration differentiation (AFPID) control algorithm and they were able to achieve a 10 dB reduction in housing vibrations at the targeted mesh harmonics. Additionally, Majumder and Tiwari (2019, 2022) designed an AVC system utilizing active magnetic bearings for a single-stage spur gearbox. They employed a feed-forward Proportional–Integral–Derivative (PID) controller. Their efforts yielded significant reductions, with geared rotor vibration levels decreased by up to 50% and an accompanying reduction in overall noise attributed to gear mesh vibrations by 4 dB.

Moreover, Zech et al. (2017, 2018a, 2019) implemented an AVC system for a high-speed planetary gearbox by utilizing piezoelectric inertial mass actuators. They constructed a test rig with an inertial actuator attached to a planetary gearbox frame and developed a control algorithm employing multiple adaptive feedforward controllers to mitigate gear meshing-induced internal forces. They were able to achieve a mean reduction of 72% of the vibration orders. Furthermore, in a previous study, the authors developed an AVC system targeting the mounting locations of an automotive transmission system housing using an add-on piezoelectric inertial piezoelectric actuator on the housing. The vibration levels are significantly suppressed within the frequency range between 1000 and 5000 Hz.

Researchers have used various types of actuators including piezoelectric, electrodynamic, magnetostrictive, electromagnetic, and dielectric elastomer actuators. However, piezoelectric actuators are commonly preferred due to their high blocking force, superior high-frequency performance, and high reliability, as evidenced in the literature. However, piezoelectric actuators with high blocking force are expensive and have high electrical capacitance, thus consuming a lot of energy, and requiring expensive and bulky power amplifiers. Moreover, the operating frequency range of the power amplifier depends on the actuator’s capacitance. The lower the capacitance, the higher the frequency operating range. For automotive gear transmission systems, most of the high amplitude vibrations and their associated harmonics tend to occur at high frequencies, therefore the targeted frequency range in this study is between 1000 and 5000 Hz.

For automotive applications, selecting an actuator with low capacitance is crucial for power efficiency. In addition, the designed actuator should be designed to be as lightweight and compact as possible. In this study, the gearbox housing is the control target, as reducing the housing vibration level would decrease the vibrations transmitted to the car body and would lead to a reduction in noise emissions. Additionally, this approach does not require high control forces, as the actuators are located far from the vibration source. Piezo shear actuators are chosen instead of regular piezo stack actuators, which have significantly lesser capacitance. Piezo shear actuators are mostly used in positioning stations. However, several studies suggested using shear actuators in composite structures as they can be integrated within the layers of the composite material. Some scholars have studied integrating shear actuators in composite structures as Senthil and Batra (2001) who developed an exact three-dimensional state space solution for the static cylindrical bending of simply supported laminated plates with embedded shear mode piezoelectric actuators.

Baillargeon and Vel (2005) performed an experimental and numerical assessment of the vibration suppression of a sandwich cantilever beam that consists of aluminum facings and a core composed of two piezoelectric shear actuators and foam. The experiments and numerical simulations show that the shear actuators could provide a significant reduction in tip acceleration and settling time.

According to the authors’ most recent literature review, Zech et al. (2018b) were the first to propose using piezoelectric shear elements for an inertial mass actuator. They built an AVC system for a planetary gearbox operating in the high-frequency range. Their work revealed that the shear actuator was capable of actively reducing the most dominant vibration order of the gearbox.

In this paper, a new concept for an integrated and energy-efficient inertial mass shear actuator is proposed. The developed actuator aims to be power-efficient and to target the local modes in the selected frequency range between 1000 and 5000 Hz. In addition, the developed inertial actuator is designed to be integrated into the housing, thus not adding extra weight to the transmission system. The proposed concept is discussed in detail in the next section. The work presented in this paper stems from the research project LIVE-I (2020).

2. Methodology

2.1. Concept

The main objective of the newly proposed actuator concept is to be implemented in an AVC system tailored for automotive gear transmission systems. An industrial gearbox from Magna Transmission Systems (7DCT300) is selected as a case study. The AVC system aims to target the sweet spots of the 7DCT300 housing. The gearbox housing is chosen to be the control object as it was found in a previous study done by Okda et al. (2022) that it does not require high control forces, therefore compact, lightweight, and economical actuators can be utilized for the active control system.

As a first step, a Finite Element (FE) modal analysis of the gearbox is performed. The FE model of the gearbox is analyzed using ANSYS software revealing the major vibration modes. The gearbox housing modes are studied until 5000 Hz, which is the operating range of the developed AVC system. This range is chosen as the gear meshing frequencies and their harmonics lie within this range.

It can be seen that specific locations on the gearbox housing are excited in most of the modes which are the targeted sweet spots for this study. One main critical point observed from the study is Position A, shown in Figure 1. Position A has two local excitation modes in the targeted frequency range, the first local bending mode exists at 2392 Hz, then the second local bending mode exists at 4231 Hz. Hence, this study aims to develop a new actuator concept that can control these two local modes effectively.

Figure 1.

Modal analysis of the gearbox housing.

Conventional Inertial Mass Actuators (IMAs) are point actuators that typically generate localized forces that are insufficient to couple with specific vibration modes dominated by distributed deformation rather than rigid body-like motion. In addition, higher-order modes, with their complex distribution of nodes and antinodes, require precise positioning of multiple actuation points of conventional inertial actuators. In addition, the actuation force pattern of conventional stack actuators does not match the spatial shape of the second bending mode well, significantly reducing suppression effectiveness.

The proposed actuator utilizes two piezoelectric shear actuators and an intermediate inertial mass to generate the required inertial force. The shear actuators are fixed to the structure from one side and fixed to an inertial mass from the other side, as shown in Figure 2. They are assembled with a small inclination angle of 15° to be securely pressed into the structure using adhesive. This proposed concept is based on the principle that the two shear actuators can be modulated through receiving dual control signals, either synchronized in phase or deliberately out of phase, thereby presenting a versatile control mechanism. The first local bending mode can be excited by supplying signals that are in phase, and the second local bending mode can be excited if the control signals are out of phase. The two excitation scenarios are also shown in Figure 2.

Figure 2.

Shear actuator excitation modes.

The newly proposed actuator is designed to offer enhanced performance compared to conventional IMAs. The proposed concept can target the two local bending modes in the targeted frequency range, while the add-on IMA can only target one mode. In addition, the shear actuator has much lesser capacitance, resulting in much lesser energy consumption. The following equations describe the relationship between the current, capacitance, voltage, and operating frequency. The maximum current required for sinusoidal operation is:

$$I_{\mathit{max}}=U_{pp}\pi fC$$

(1)

and the maximum power necessary for this mode of operation is given by:

$$P_{\mathit{max}}=U_{pp}^2\pi f_{\mathit{max}}C$$

(2)

where I_max is the maximum amplifier current, f_max is the maximum operating frequency, C is the piezo actuator capacitance, and U _pp is the peak–peak drive voltage. It can be observed that the consumed power relies on the capacitance of the actuator.

Moreover, low capacitance actuators lead to a better power amplifier linear performance at high frequencies. For example, Figure 3 shows a chart of a commercial power amplifier EPA-104 from Piezo.com. It is shown that the performance of the amplifier resembled by the supplied voltage and the operating frequency range depends on the capacitance of the actuator. Therefore, actuators with low capacitance have better linear voltage behavior at high frequencies.

Figure 3.

Peak voltage delivered to a capacitive load at a 200 mA peak current rating, as a function of the operating frequency of the EPA-104 power amplifier from Piezo.com.

The integration of the actuation mechanism offers advantages such as reduced size, reduced weight, and better functionality. Compact-size shear actuators can be integrated within the thickness of the gearbox housing by introducing a cutout in the housing to fit the inertial shear actuator. However, it should be protected from the internal lubrication and the external hits and dust. A concept for integrating the inertial shear actuator is presented in Figure 4.

Figure 4.

Integration of the shear actuator schematic.

In order to validate the concept of the dual-mode actuator, it is tested on a basic plate structure. The plate dimensions are designed to have bending modes in the same frequency range as the bending modes of the local position A in the gearbox housing. In addition, the plate material is selected to be similar to the housing, which is composed of an aluminum alloy. The specifications of the developed actuators are presented in Table 1.

Table 1.

Inertial shear actuator specifications.

Specification	Value
Operating voltage range (V)	−250 to 250
Operating temperature range (°C)	−20 to 85
Inertial shear actuator weight (g)	32.6
Stack dimensions (mm)	5 × 5 × 3.5
Inertial actuator dimensions (mm)	19.5 × 7.8 × 5.5
Electrical capacitance (nF)	1.4 ± 20%
Max. shear load (N)	50
Displacement (Free stroke) (μm)	1 ± 30%
Stack axial resonant frequency (kHz)	330

2.2. Developed model

To evaluate the novel concept, a small test setup is designed to accommodate the plate and the shear actuators as well as an additional exciting actuator. Subsequently, a model is created to simulate the new concept of the inertial shear actuator, to ensure that both the two bending modes of the plate can be excited and that the actuators are powerful enough to counteract the induced vibrations. A FE modal analysis is conducted to select a proper excitation point to excite the two bending modes and to ensure that the bending modes are within the same range of the gearbox housing. The modal analysis of the test setup, shown in Figure 5, shows that the first two bending modes occur at approximately 2940 Hz and 4787 Hz, which are in the same range as the local bending modes of the gearbox housing.

Figure 5.

Modal analysis of test setup.

A harmonic analysis is then conducted to excite the plate at the excitation point. Measuring point 1 is selected near the center to detect the first bending mode and measuring point 2 is selected at the tip to detect the second bending mode. The excitation point and the acceleration measuring points are also shown in Figure 5.

The developed model is performed using Matlab Simulink and it consists of several modules which are the mechanical structure, the shear actuators, the inertial mass, and the controller. In addition, other modules can be introduced to the model in future studies as the sensor module and the power amplifier module. However, they are not accounted for in this paper as the primary focus is on the actuator.

2.2.1. Mechanical structure modeling

Mechanical structures have the most significant contribution to the dynamic properties. To obtain a numerical model of the mechanical structure, the FEA method is selected and is performed using ANSYS software. However, the FE model has a high number of Degrees Of Freedom (DOF), which makes the model complicated. Therefore, the FE model is reduced to the first 22 modes, which occur in the frequency range between 0 and 12.4 kHz, which is beyond the targeted frequency range of 5000 Hz. The reduced model is then converted to a state space model with 44 states, having three input force points, which are the excitation point and the two shear actuator influence points. In addition, there are 12 output points which are the displacements, velocities, and accelerations of the shear actuator/plate interaction points and the two measuring points. The resulting state space equations are described as follows:

$$\dot{\tilde{x}}(t)=\tilde{A}\tilde{x}(t)+\tilde{B}u(t)$$

(3)

$$y(t)=\tilde{C}\tilde{x}(t)+Du(t)$$

(4)

$$\begin{array}{ll}\hfill \tilde{A}& =\left[\begin{array}{cc}\hfill [0]\hfill & \hfill [I]\hfill \\ \hfill [-diag({\omega }^2)]\hfill & \hfill [-diag(2\zeta \omega )]\hfill \end{array}\right];\hfill \\ \hfill \tilde{B}& =\left[\begin{array}{c}\hfill [0]\hfill \\ \hfill {[\varphi ]}^T[F_u]\hfill \end{array}\right];\hfill \\ \hfill \tilde{C}& =\left[\begin{array}{cc}\hfill [D_u][\varphi ]\hfill & \hfill [0]\hfill \\ \hfill [0]\hfill & \hfill [D_u][\varphi ]\hfill \\ \hfill [D_u][\varphi ][-diag({\omega }^2)]\hfill & \hfill [D_u][\varphi ][-diag(2\zeta \omega )]\hfill \end{array}\right];\hfill \\ \hfill \tilde{D}& =\left[\begin{array}{c}\hfill [0]\hfill \\ \hfill [0]\hfill \\ \hfill [D_u][\varphi ]{[\varphi ]}^T[F_u]\hfill \end{array}\right];\hfill \end{array}$$

where $\tilde{x}$ is the modal state vector, y is the output vector, u is the input or control vector, $\tilde{A}$ is the modal system matrix, $\tilde{B}$ is the modal input matrix, $\tilde{C}$ is the modal output matrix, and $\tilde{D}$ is the feed-through matrix. The modal state vector $\tilde{x}$ has a dimension of 2n, where n is the number of modes, $\tilde{A}$ matrix must be a 2n × 2n matrix. $\tilde{B}$ matrix must be a 2n × m matrix, where m is the number of inputs. $\tilde{C}$ matrix must be a 3r × 2n matrix, where r is the number of outputs. D must be a 3r × m matrix. The term ζ refers to the structural modal damping. The ϕ columns correspond to the n eigenvectors and ω refers to the n eigenvalues. F _u is a unit force matrix, it has 1 at the degrees of freedom where input forces are active and 0 elsewhere. D _u is a unit displacement matrix, it has 1 on degrees of freedom where output is requested and 0 elsewhere. The modal damping is estimated and added to the state space matrices.

2.2.2. Shear actuator modeling

The modeling approach of Herold et al. (2009, 2011) is used in the model, which provides alternating impedance and admittance formulations between the mechanical structure and the actuators. The coupling between the mechanical structure and the shear actuator is considered at discrete points. An admittance description is chosen for the mechanical structure of the plate and the piezoelectric shear actuator is formulated accordingly in an impedance representation. This definition allows also for implementing nonlinear behavior for both formulations. The input variables of the shear actuator module are the voltage of the power amplifier and the velocity of the connection points. The output variable of the actuator module is the force. On the other hand, in the plate mechanical structure, the force is the input variable while the velocity of the actuator connection points is the output.

Furthermore, the piezoelectric shear actuator modules are implemented using the AdaptroSim™ toolbox, which is developed at the Fraunhofer LBF (AdaptroSim, 2023). An analytic formulation is used in the tool to describe the actuator. The shear actuator module comprises the mechanical properties of the actuator (stiffness and viscous damping) as well as an electro-mechanical coupling between the force and the applied voltage. For the modeling approach, it is assumed that the electrical interaction between the actuator and amplifier can be neglected at this stage. This assumes an infinite electrical source strength powering the actuator. For this reason, modeling the power amplifier is not required here, but can be beneficial for later and more refined development steps.

The model’s inputs are the velocities v₁ and v₂ at both ends of the actuator as well as the driving voltage U. The block’s outputs are the resulting forces F₁ and F₂ at the corresponding positions. Thus, the transducer is described in terms of impedance in the mechanical domain. A schematic of the AdaptroSim actuator module is shown in Figure 6 and the force actuator equation is described by:

$$F_1=-F_2=\left(\frac{k}{s}+d\right)(v_2-v_1)+\alpha U$$

(5)

Figure 6.

Schematic for the actuator model.

where U is the actuator input voltage, k and d are the stiffness and damping coefficient of the support and α denotes the coupling factor.

The shear actuator stiffness and the coupling factor between the actuator input voltage and output force are determined experimentally by attaching a determined mass to the shear actuator and measuring the mass tip acceleration using a laser vibrometer of type (PSV-500-3D) from Polytec. To acquire the actuator stiffness, the actuator is supplied with a sine sweep signal, and its natural frequency is determined f _n , then the stiffness value k is calculated from equation (6).

$$f_n=\frac{1}{2\pi }\sqrt{\frac{k}{m}}$$

(6)

The coupling value is defined by applying a known voltage to the actuator and subsequently deriving the actuator force based on the measured acceleration, considering the known mass. The testing setup is shown in Figure 7. The damping parameter d is estimated for the shear actuator module. The excitation force is modeled as an ideal sinusoidal force supplied to the excitation point.

Figure 7.

Experimental setup for measuring actuator stiffness.

2.2.3. Control algorithm

For the controller block, the widely used FxLMS algorithm is implemented in the Simulink model to control the shear actuators and reduce the acceleration at the measuring points. The control voltage is limited to 120 V due to the power amplifier limitations.

The total vibration signal measured on the plate is denoted by $\widehat{\mathbf{q}}(\mathbf{t})$ . According to the dynamic equations of the system, the vibration signal can be described as:

$$\widehat{\mathbf{q}}(\mathbf{t})=\widehat{\mathbf{d}}(\mathbf{t})+\widehat{\mathbf{H}}\widehat{{\mathbf{U}}_{\mathbf{a}}}(\mathbf{t})$$

(7)

$\widehat{\mathbf{d}}(\mathbf{t})$ is the vibration occurring at the mounting location due to the primary excitation when no control forces are applied. The control voltage calculated by FxLMS algorithm, $\widehat{{\mathbf{U}}_{\mathbf{a}}}(\mathbf{t})$ is fed to the associated power amplifier of the shear actuator to suppress the vibration. The vibration path from the center of the plate where the shear actuator is installed to the deformation of a particular is the secondary path indicated by $\widehat{\mathbf{H}}$ .

The frequency-dependent transfer behavior of the secondary path, that is, denoted by $\widehat{\mathbf{H}}=H_j{e}^{i{\varphi }_j}$ . The term ϕ _j is the phase lag experienced by the signal that travels from the controller to the shear actuator location and then to the position of maximum deformation of the mode to cancel it. This phase lag is a combination of mechanical and electrical delays. For a simple plate arrangement of the shear actuator set-up, the secondary path phase is estimated from the finite-element model for the two vibrating modes shown in Figure 5.

The schematic of the FxLMS algorithm is given in Figure 8, where μ is the learning rate.

Figure 8.

Schematic of the FxLMS Algorithm used.

For a sinusoidal excitation of the two vibrating modes, the resultant control force would also be sinusoidal, that is, a combination of sine and cosine components, given by equation (8), to its associated power amplifier.

$$\widehat{{\mathbf{U}}_{\mathbf{a}}}(\mathbf{t})=(U_{a_j}^c+i{U}_{a_j}^s)e^{i{\omega }_{j}t}$$

(8)

Substituting complex quantities $\widehat{\mathbf{H}}$ and $\widehat{{\mathbf{U}}_{\mathbf{a}}}(\mathbf{t})$ , equation (7) is rewritten as follows:

$$\widehat{\mathbf{q}}(\mathbf{t})=\widehat{\mathbf{d}}(\mathbf{t})+H_j(U_{a_j}^c+i{U}_{a_j}^s)e^{i({\omega }_{j}t+{\varphi }_j)}$$

(9)

The sensor measures only the real part of the signal in practical applications. From equation (9), retaining the real components gives the mathematical expression for the resulting vibration signal recorded as a sensor function of all the physical phenomena in the designed active vibration control system.

$$q(t)=d(t)+H_j(U_{a_j}^c\cos ({\omega }_{j}t+{\varphi }_j)+U_{a_j}^s\sin ({\omega }_{j}t+{\varphi }_j))$$

(10)

The control voltage components $U_{a_j}^c$ and $U_{a_j}^s$ will be adapted in an iterative manner using FxLMS at the designated vibrating modes denoted by ω _j , where j = 1 is the first bending mode and j = 2 is the second bending mode.

$$\begin{array}{cc}\hfill U_a(t)& =U_{a_j}^c\cos ({\omega }_{j}t+{\varphi }_j)+U_{a_j}^s\sin ({\omega }_{j}t+{\varphi }_j)\hfill \\ \hfill & =\left[\begin{array}{cc}\hfill \cos ({\omega }_{j}t+{\varphi }_j)\hfill & \hfill \sin ({\omega }_{j}t+{\varphi }_j)\hfill \end{array}\right]\left\{\begin{array}{c}\hfill U_{a_j}^c\hfill \\ \hfill U_{a_j}^s\hfill \end{array}\right\}\hfill \\ \hfill & =X_j(t)\left\{\begin{array}{c}\hfill U_{a_j}^c\hfill \\ \hfill U_{a_j}^s\hfill \end{array}\right\}\hfill \end{array}$$

(11)

The reference signal correlated with the input excitation frequency along with the frequency-dependent lag can be generated in vector form as: $X_j(t)=\left[\begin{array}{cc}\hfill \cos ({\omega }_{j}t+{\varphi }_j)\hfill & \hfill \sin ({\omega }_{j}t+{\varphi }_j)\hfill \end{array}\right]$ .

The objective of the control algorithm is to minimize the instantaneous squared quantity of the vibration signal, that is, ζ = (q(t))². The gradient descent update is employed to update the coefficients of U _a (t) to minimize the vibration error signal. The gradient of the vibration signal for each of the coefficients of U _a (t) given by equation (10) is given as:

$$\begin{array}{cc}\hfill ▿\zeta =\frac{d\zeta }{d(U_a(t))}& =2q(t)\left\{\begin{array}{c}\hfill \frac{d(q(t))}{d(U_{a_j}^c)}\hfill \\ \hfill \frac{d(q(t))}{d(U_{a_j}^s)}\hfill \end{array}\right\}\hfill \\ \hfill & =2q(t)\left\{\begin{array}{c}\hfill H_j\cos ({\omega }_{j}t+{\varphi }_j)\hfill \\ \hfill H_j\sin ({\omega }_{j}t+{\varphi }_j)\hfill \end{array}\right\}\hfill \end{array}$$

(12)

The coefficients ${\left[\begin{array}{cc}\hfill U_{a_j}^c\hfill & \hfill U_{a_j}^s\hfill \end{array}\right]}^T$ are initialized to zero at the beginning of the test. According to the algorithm, they are updated for each time step Δt as follows:

$$\begin{array}{cc}\hfill {\left\{\begin{array}{c}\hfill U_{a_j}^c\hfill \\ \hfill U_{a_j}^s\hfill \end{array}\right\}}_{t+\mathrm{\Delta }t}& ={\left\{\begin{array}{c}\hfill U_{a_j}^c\hfill \\ \hfill U_{a_j}^s\hfill \end{array}\right\}}_t-\frac{1}{2}\mu \mathrm{\Delta }t▿\zeta \hfill \\ \hfill & ={\left\{\begin{array}{c}\hfill U_{a_j}^c\hfill \\ \hfill U_{a_j}^s\hfill \end{array}\right\}}_t-\mu \mathrm{\Delta }t\left\{\begin{array}{c}\hfill H_j\cos ({\omega }_{j}t+{\varphi }_j)\hfill \\ \hfill H_j\sin ({\omega }_{j}t+{\varphi }_j)\hfill \end{array}\right\}q(t)\hfill \\ \hfill & ={\left\{\begin{array}{c}\hfill U_{a_j}^c\hfill \\ \hfill U_{a_j}^s\hfill \end{array}\right\}}_t-\mu \mathrm{\Delta }t{H}_j{X}_j^T(t)\hfill \end{array}$$

(13)

The schematic of the developed model is shown in Figure 9. In the Simulink model schematic, the state space model is shown along with the actuator modules, the inertial mass as well as the controller module. The power amplifier is just modeled as a constant gain.

Figure 9.

Schematic of the Simulink model.

The excitation adopted in the simulation model of the active system is an ideal sine force excitation. Upon constructing the model, two excitation signals are given at the excitation point sequentially. The acceleration levels of the control points are measured and the resulting vibration signal is captured. The output of the control signals is adapted in real time and is sent to the shear actuators from the controller. The controller is modeled to work after 0.15 seconds. For the first excitation signal, the two signals are given in phase, and for the second excitation signal, the two signals are given out of phase. Figure 10 illustrates the acceleration levels at the measuring points for both excitation signals, comparing the conditions with and without control. It can be seen that the actuators can effectively eliminate vibrations, achieving a reduction of 60 dB for the first mode and 50 dB for the second mode.

Figure 10.

Simulink model acceleration levels before and after control.

2.3. Experimental setup

Upon the completion of model construction and its subsequent successful evaluation, an experimental setup is developed to validate the newly developed concept. As previously mentioned, the setup aims to introduce an excitation signal to the plate that can excite both the first and second bending modes and cancel these vibrations using the developed shear actuators. The setup utilized an aluminum plate and a mechanical structure to hold all of the components. Moreover, two compact piezo shear actuators (P-121.01) from Physik Instrumente (PI) are installed to counteract the vibrations, this shear actuator is chosen due to its small size compared to the thickness of the plate. The size of the actuators is shown in Figure 11, the shear actuator has a capacitance of 1.4 nF.

Figure 11.

The developed inertial shear actuator size.

In addition, Three Micro Fiber Composite (MFC) sensors are used to measure the vibration levels on the plate (Smart Material GMBH, 2021). MFC 1 is used at measuring position (1) and MFC 2 is used at measuring position (2) as shown in Figure 5. An additional sensor (MFC 3) is used at the other tip of the plate to make sure that the second bending mode is controlled and that both readings on the two tips are being reduced. Only one measuring point per mode is chosen for the controller, which is MFC 1 and MFC 2 as the test setup is symmetrical.

MFCs are smart composites comprising alternating layers of piezoelectric matrix and metallic electrodes. The piezoelectric materials generate charges due to deformation and act as a sensitive yet effective vibration sensor. They are adaptable to various surfaces and can be integrated into structures. That means that this surface-conformable sheet can be mounted easily onto various structures, which suits the complex geometry of the housing. In addition, they have a wide bandwidth of operation up to a frequency range of 1 MHz and low cost. Sodano et al. (2004) used MFCs for vibration suppression and structural health monitoring. They proved that an MFC could be used as both a sensor and actuator, then used them to determine the modal parameters of an inflatable structure.

The control algorithm is deployed in real-time on an economical and fast microcontroller, which is the Teensy (2023). The microcontroller includes a 600 MHz ARM-M7 processor. The microcontroller also performs relevant data acquisition and processing to calculate the relevant control output for the actuator. The control algorithm described in the previous section is applied to actively control the vibrations of the two main modes of the plate set-up.

Furthermore, a compact IMA is used to excite the system, the exciting actuator is screwed to the center of the base. The voltage supplied to the exciting IMA is adjusted to achieve a level of vibration that has the same order as the gearbox housing. A dynamic force sensor (9001A) from Kistler is used to measure the input excitation force. This force value was used in the model as the excitation force. The test setup is assembled successfully with the integrated shear actuators and it is shown in Figure 12.

Figure 12.

Test setup of the shear actuator.

3. Results and discussion

A sine chirp signal is supplied to the plate at the excitation point by the exciting IMA to determine the bending modes. It is found that the first bending mode occurs at 2820 Hz and the second bending mode occurs at 4850 Hz, which shows a good correlation with the FE model. Then the exciting IMA is supplied with an excitation sinusoidal signal, the acceleration levels are measured after calibrating the MFCs with an accelerometer, and then other control signals are supplied by the controller to cancel out the excitation signal.

The experimental results are compared to the results obtained from the model, and both the time and frequency domain signals of both modes before and after control are shown in Figures 13–15. It is shown that for the first bending mode, MFC 1 is observed and it is shown that the vibration levels are reduced significantly, and a reduction of approximately 36 dB is achieved. For the second bending mode, MFC 2 and MFC 3 are observed and a reduction of approximately 41 dB is achieved at the position of MFC 2 which is the control point. Moreover, the acceleration level at MFC 3 is also reduced which means that the second bending mode is controlled successfully. However, the amount of excitation at MFC 3 exhibits a higher vibration magnitude compared to MFC 2. The reduction in the acceleration level at MFC 3 is found to be 9 dB, which is less than MFC 2. This can be attributed to some imperfections that lead to non-ideal symmetry in the mechanical setup or the excitation mechanism of the plate. Furthermore, the controller converged within 0.2 seconds of activation for both modes.

Figure 13.

MFC1 sensor signal without and with activated control of the first mode. (a) The top graph shows the time domain sensor signal. (b) The bottom graph shows the frequency domain sensor signal.

Figure 14.

MFC2 sensor signal without and with activated control of the second mode. (a) The top graph shows the time domain sensor signal. (b) The bottom graph shows the frequency domain sensor signal.

Figure 15.

MFC3 sensor signal without and with activated control of the second mode. (a) The top graph shows the time domain sensor signal. (b) The bottom graph shows the frequency domain sensor signal.

4. Conclusion

A new concept is presented to integrate a piezoelectric shear actuator in the housing of an automotive gearbox. The newly developed actuator offers several advantages as reduced weight, lower power consumption, high-frequency operation, and improved housing accommodation. In addition, it surpasses the functionality of the traditional IMA by targeting two local modes within the operating frequency range, whereas the add-on IMA can only address one mode. Moreover, the developed actuator consumes much less power due to its low capacitance and can operate at a higher frequency range. A model is created to simulate the new concept of the shear actuator. The model is built in Matlab Simulink and it is seen that the proposed actuator can suppress the vibrations successfully. An experimental setup is developed to validate the newly developed concept on a simple plate structure. MFC sensors are used to capture the control signals, and a controller using the FxLMS algorithm is used to control the vibrations of the observed points. An additional compact inertial actuator is used to excite both the first and second bending modes of the plate and the shear actuators were able to efficiently cancel these vibrations. The first mode steady state vibration level is reduced by approximately 36 dB, while the second bending mode steady state vibration level is reduced by 41 dB at the control point and reduced by 9 dB at the other observing point. Nevertheless, a substantial reduction in the second mode is still achieved. Subsequent to the validation of the AVC system on the plate structure it is evident that the AVC system, featuring the innovative actuator concept, has displayed both success and promise.

The upcoming research phase aims to evaluate the AVC system’s performance on the gearbox housing. Additionally, there’s potential for deeper exploration into sensor dynamics and a thorough assessment of power amplifier performance. These investigations aim to enrich the existing model for enhanced functionality and comprehensiveness.